An open question in \emph{Imprecise Probabilistic Machine Learning} is how to empirically derive a credal region (i.e., a closed and convex family of probabilities on the output space) from the available data, without any prior knowledge or assumption. In classification problems, credal regions are a tool that is able to provide provable guarantees under realistic assumptions by characterizing the uncertainty about the distribution of the labels. Building on previous work, we show that credal regions can be directly constructed using conformal methods. This allows us to provide a novel extension of classical conformal prediction to problems with ambiguous ground truth, that is, when the exact labels for given inputs are not exactly known. The resulting construction enjoys desirable practical and theoretical properties: (i) conformal coverage guarantees, (ii) smaller prediction sets (compared to classical conformal prediction regions) and (iii) disentanglement of uncertainty sources (epistemic, aleatoric). We empirically verify our findings on both synthetic and real datasets.
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In this note we show examples of total Boolean functions that depend on $n$ variables and have spectral sensitivity $\Theta(\sqrt{\log n})$, which is asymptotically minimal.
Neural operators effectively solve PDE problems from data without knowing the explicit equations, which learn the map from the input sequences of observed samples to the predicted values. Most existing works build the model in the original geometric space, leading to high computational costs when the number of sample points is large. We present the Latent Neural Operator (LNO) solving PDEs in the latent space. In particular, we first propose Physics-Cross-Attention (PhCA) transforming representation from the geometric space to the latent space, then learn the operator in the latent space, and finally recover the real-world geometric space via the inverse PhCA map. Our model retains flexibility that can decode values in any position not limited to locations defined in the training set, and therefore can naturally perform interpolation and extrapolation tasks particularly useful for inverse problems. Moreover, the proposed LNO improves both prediction accuracy and computational efficiency. Experiments show that LNO reduces the GPU memory by 50%, speeds up training 1.8 times, and reaches state-of-the-art accuracy on four out of six benchmarks for forward problems and a benchmark for inverse problem. Code is available at //github.com/L-I-M-I-T/LatentNeuralOperator.
Extracting time-varying latent variables from computational cognitive models is a key step in model-based neural analysis, which aims to understand the neural correlates of cognitive processes. However, existing methods only allow researchers to infer latent variables that explain subjects' behavior in a relatively small class of cognitive models. For example, a broad class of relevant cognitive models with analytically intractable likelihood is currently out of reach from standard techniques, based on Maximum a Posteriori parameter estimation. Here, we present an approach that extends neural Bayes estimation to learn a direct mapping between experimental data and the targeted latent variable space using recurrent neural networks and simulated datasets. We show that our approach achieves competitive performance in inferring latent variable sequences in both tractable and intractable models. Furthermore, the approach is generalizable across different computational models and is adaptable for both continuous and discrete latent spaces. We then demonstrate its applicability in real world datasets. Our work underscores that combining recurrent neural networks and simulation-based inference to identify latent variable sequences can enable researchers to access a wider class of cognitive models for model-based neural analyses, and thus test a broader set of theories.
Concurrent computation and communication (C3) is a pervasive paradigm in ML and other domains, making its performance optimization crucial. In this paper, we carefully characterize C3 in ML on GPUs, which are most widely deployed for ML training and inference. We observe that while C3 leads to performance uplifts, the uplifts are far lower than ideal speedups (serial computation and communication versus maximum of computation or communication; all times from isolated executions). C3 on average achieves only 21% of ideal speedup, this is due to known challenges of compute and memory interference between concurrent GPU kernels (that is, sharing of GPU's compute units, caches and HBM). To attain better performance for C3, first, we evaluate dual strategies of schedule prioritization and careful resource partitioning of compute units on GPUs to push performance attained with C3 (on average 42% of ideal speedup). We also provide heuristics that can guide a runtime while employing these strategies. To further enhance C3 performance, we propose to mitigate C3 interference by offloading communication tasks to the GPU's DMA engines. To this end, we build Concurrent Communication CoLlectives (ConCCL) proof-of-concepts that harness DMA engines for communication. We show how ConCCL considerably closes the gap between realized and ideal speedup for C3 (on average 72% of ideal speedup is realized, up to 1.67x speedup). Overall, our work makes a strong case for GPU DMA engine advancements to better support C3 on GPUs.
Schr\"{o}dinger Bridges (SB) are diffusion processes that steer, in finite time, a given initial distribution to another final one while minimizing a suitable cost functional. Although various methods for computing SBs have recently been proposed in the literature, most of these approaches require computationally expensive training schemes, even for solving low-dimensional problems. In this work, we propose an analytic parametrization of a set of feasible policies for steering the distribution of a dynamical system from one Gaussian Mixture Model (GMM) to another. Instead of relying on standard non-convex optimization techniques, the optimal policy within the set can be approximated as the solution of a low-dimensional linear program whose dimension scales linearly with the number of components in each mixture. Furthermore, our method generalizes naturally to more general classes of dynamical systems such as controllable Linear Time-Varying systems that cannot currently be solved using traditional neural SB approaches. We showcase the potential of this approach in low-to-moderate dimensional problems such as image-to-image translation in the latent space of an autoencoder, and various other examples. We also benchmark our approach on an Entropic Optimal Transport (EOT) problem and show that it outperforms state-of-the-art methods in cases where the boundary distributions are mixture models while requiring virtually no training.
Forecasting relations between entities is paramount in the current era of data and AI. However, it is often overlooked that real-world relationships are inherently directional, involve more than two entities, and can change with time. In this paper, we provide a comprehensive solution to the problem of forecasting directional relations in a general setting, where relations are higher-order, i.e., directed hyperedges in a hypergraph. This problem has not been previously explored in the existing literature. The primary challenge in solving this problem is that the number of possible hyperedges is exponential in the number of nodes at each event time. To overcome this, we propose a sequential generative approach that segments the forecasting process into multiple stages, each contingent upon the preceding stages, thereby reducing the search space involved in predictions of hyperedges. The first stage involves a temporal point process-based node event forecasting module that identifies the subset of nodes involved in an event. The second stage is a candidate generation module that predicts hyperedge sizes and adjacency vectors for nodes observing events. The final stage is a directed hyperedge predictor that identifies the truth by searching over the set of candidate hyperedges. To validate the effectiveness of our model, we compiled five datasets and conducted an extensive empirical study to assess each downstream task. Our proposed method achieves a performance gain of 32\% and 41\% compared to the state-of-the-art pairwise and hyperedge event forecasting models, respectively, for the event type prediction.
Graphs are important data representations for describing objects and their relationships, which appear in a wide diversity of real-world scenarios. As one of a critical problem in this area, graph generation considers learning the distributions of given graphs and generating more novel graphs. Owing to their wide range of applications, generative models for graphs, which have a rich history, however, are traditionally hand-crafted and only capable of modeling a few statistical properties of graphs. Recent advances in deep generative models for graph generation is an important step towards improving the fidelity of generated graphs and paves the way for new kinds of applications. This article provides an extensive overview of the literature in the field of deep generative models for graph generation. Firstly, the formal definition of deep generative models for the graph generation and the preliminary knowledge are provided. Secondly, taxonomies of deep generative models for both unconditional and conditional graph generation are proposed respectively; the existing works of each are compared and analyzed. After that, an overview of the evaluation metrics in this specific domain is provided. Finally, the applications that deep graph generation enables are summarized and five promising future research directions are highlighted.
Graph Convolutional Networks (GCNs) have been widely applied in various fields due to their significant power on processing graph-structured data. Typical GCN and its variants work under a homophily assumption (i.e., nodes with same class are prone to connect to each other), while ignoring the heterophily which exists in many real-world networks (i.e., nodes with different classes tend to form edges). Existing methods deal with heterophily by mainly aggregating higher-order neighborhoods or combing the immediate representations, which leads to noise and irrelevant information in the result. But these methods did not change the propagation mechanism which works under homophily assumption (that is a fundamental part of GCNs). This makes it difficult to distinguish the representation of nodes from different classes. To address this problem, in this paper we design a novel propagation mechanism, which can automatically change the propagation and aggregation process according to homophily or heterophily between node pairs. To adaptively learn the propagation process, we introduce two measurements of homophily degree between node pairs, which is learned based on topological and attribute information, respectively. Then we incorporate the learnable homophily degree into the graph convolution framework, which is trained in an end-to-end schema, enabling it to go beyond the assumption of homophily. More importantly, we theoretically prove that our model can constrain the similarity of representations between nodes according to their homophily degree. Experiments on seven real-world datasets demonstrate that this new approach outperforms the state-of-the-art methods under heterophily or low homophily, and gains competitive performance under homophily.
Incompleteness is a common problem for existing knowledge graphs (KGs), and the completion of KG which aims to predict links between entities is challenging. Most existing KG completion methods only consider the direct relation between nodes and ignore the relation paths which contain useful information for link prediction. Recently, a few methods take relation paths into consideration but pay less attention to the order of relations in paths which is important for reasoning. In addition, these path-based models always ignore nonlinear contributions of path features for link prediction. To solve these problems, we propose a novel KG completion method named OPTransE. Instead of embedding both entities of a relation into the same latent space as in previous methods, we project the head entity and the tail entity of each relation into different spaces to guarantee the order of relations in the path. Meanwhile, we adopt a pooling strategy to extract nonlinear and complex features of different paths to further improve the performance of link prediction. Experimental results on two benchmark datasets show that the proposed model OPTransE performs better than state-of-the-art methods.
Image segmentation is still an open problem especially when intensities of the interested objects are overlapped due to the presence of intensity inhomogeneity (also known as bias field). To segment images with intensity inhomogeneities, a bias correction embedded level set model is proposed where Inhomogeneities are Estimated by Orthogonal Primary Functions (IEOPF). In the proposed model, the smoothly varying bias is estimated by a linear combination of a given set of orthogonal primary functions. An inhomogeneous intensity clustering energy is then defined and membership functions of the clusters described by the level set function are introduced to rewrite the energy as a data term of the proposed model. Similar to popular level set methods, a regularization term and an arc length term are also included to regularize and smooth the level set function, respectively. The proposed model is then extended to multichannel and multiphase patterns to segment colourful images and images with multiple objects, respectively. It has been extensively tested on both synthetic and real images that are widely used in the literature and public BrainWeb and IBSR datasets. Experimental results and comparison with state-of-the-art methods demonstrate that advantages of the proposed model in terms of bias correction and segmentation accuracy.
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